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June  2016, 11(2): 203-222. doi: 10.3934/nhm.2016.11.203

Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws

Boris Andreianov 1, and  Mohamed Karimou Gazibo 2,

1. 

Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 25030 Besançon Cedex
2. 

Université Abdou Moumouni de Niamey, École Normale Supérieure, Departement de Mathématiques, BP: 10963 Niamey, Niger

Received  May 2015 Revised  October 2015 Published  March 2016

We revisit the Cauchy-Dirichlet problem for degenerate parabolic scalar conservation laws. We suggest a new notion of strong entropy solution. It gives a straightforward explicit characterization of the boundary values of the solution and of the flux, and leads to a concise and natural uniqueness proof, compared to the one of the fundamental work [J. Carrillo, Arch. Ration. Mech. Anal., 1999]. Moreover, general dissipative boundary conditions can be studied in the same framework. The definition makes sense under the specific weak trace-regularity assumption. Despite the lack of evidence that generic solutions are trace-regular (especially in space dimension larger than one), the strong entropy formulation may be useful for modeling and numerical purposes.
Keywords: weakly trace-regular solutions, Degenerate parabolic-hyperbolic equation, strong entropy formulation, uniqueness proof., Dirichlet boundary condition.
Mathematics Subject Classification: Primary: 35M13; Secondary: 35L0.
Citation: Boris Andreianov, Mohamed Karimou Gazibo. Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws. Networks & Heterogeneous Media, 2016, 11 (2) : 203-222. doi: 10.3934/nhm.2016.11.203
References:
[1]

J. Aleksic and D. Mitrovic, Strong traces for averaged solutions of heterogeneous ultra-parabolic transport equations,, J. Hyperb. Differ. Equ., 10 (2013), 659.  doi: 10.1142/S0219891613500239.  Google Scholar

[2]

K. Ammar, P. Wittbold and J. Carrillo, Scalar conservation laws with general boundary condition and continuous flux function,, J. Differ. Equ., 228 (2006), 111.  doi: 10.1016/j.jde.2006.05.002.  Google Scholar

[3]

B. Andreianov, M. Bendahmane and K. H. Karlsen, Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations,, J. Hyperb. Differ. Equ., 7 (2010), 1.  doi: 10.1142/S0219891610002062.  Google Scholar

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B. Andreianov and F. Bouhsiss, Uniqueness for an elliptic-parabolic problem with Neumann boundary condition,, J. Evol. Equ., 4 (2004), 273.  doi: 10.1007/s00028-004-0143-1.  Google Scholar

[5]

B. Andreianov and M. Karimou Gazibo, Entropy formulation of degenerate parabolic equation with zero-flux boundary condition,, Z. Angew. Math. Phys., 64 (2013), 1471.  doi: 10.1007/s00033-012-0297-6.  Google Scholar

[6]

B. Andreianov and M. Karimou Gazibo, Convergence of finite volume scheme for degenerate parabolic problem with zero flux boundary condition，, Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects, 77 (2014), 303.  doi: 10.1007/978-3-319-05684-5_29.  Google Scholar

[7]

B. Andreianov and K. Shibi, Scalar conservation laws with nonlinear boundary conditions,, C. R. Acad. Paris Ser. I Math., 345 (2007), 431.  doi: 10.1016/j.crma.2007.09.008.  Google Scholar

[8]

B. Andreianov and K. Sbihi, Well-posedness of general boundary-value problems for scalar conservation laws,, Trans. AMS, 367 (2015), 3763.  doi: 10.1090/S0002-9947-2015-05988-1.  Google Scholar

[9]

C. Bardos, A.-Y. LeRoux and J.-C. Nédélec, First order quasilinear equations with boundary conditions,, Comm. PDE, 4 (1979), 1017.  doi: 10.1080/03605307908820117.  Google Scholar

[10]

Ph. Bénilan, Equations D'évolution dans un Espace de Banach Quelconque et Applications,, Thèse d'état, (1972).   Google Scholar

[11]

Ph. Bénilan, M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces,, Preprint book., ().   Google Scholar

[12]

Ph. Bénilan, J. Carrillo and P. Wittbold, Renormalized entropy solutions of scalar conservation laws,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 313.   Google Scholar

[13]

R. Bürger, H. Frid and K.H. Karlsen, On the well-posedness of entropy solution to conservation laws with a zero-flux boundary condition,, J. Math. Anal. Appl., 326 (2007), 108.  doi: 10.1016/j.jmaa.2006.02.072.  Google Scholar

[14]

R. Bürger, H. Frid and K. H. Karlsen, On a free boundary problem for a strongly degenerate quasilinear parabolic equation with an application to a model of presssure filtration,, SIAM J. Math. Anal., 34 (2003), 611.   Google Scholar

[15]

C. Cancès, Th. Gallouët and A. Porretta, Two-phase flows involving capillary barriers in heterogeneous porous media,, Interfaces Free Bound, 11 (2009), 239.  doi: 10.4171/IFB/210.  Google Scholar

[16]

J. Carrillo, Entropy solutions for nonlinear degenerate problems,, Arch. Ration. Mech. Anal., 147 (1999), 269.  doi: 10.1007/s002050050152.  Google Scholar

[17]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws,, Arch. Ration. Mech. Anal., 147 (1999), 89.  doi: 10.1007/s002050050146.  Google Scholar

[18]

R. Colombo and E. Rossi, Rigorous estimates on balance laws in bounded domains,, Acta Mathematica Sci., 35 (2015), 906.  doi: 10.1016/S0252-9602(15)30028-X.  Google Scholar

[19]

F. Dubois and Ph. LeFloch, Boundary condition for nonlinear hyperbolic conservation laws,, J. Differ. Equ., 71 (1988), 93.  doi: 10.1016/0022-0396(88)90040-X.  Google Scholar

[20]

S. Evje and K. H. Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations,, SIAM J. Numer. Anal., 37 (2000), 1838.  doi: 10.1137/S0036142998336138.  Google Scholar

[21]

G. Gagneux and M. Madaune-Tort, Analyse Mathématique de Modeles Nonlinéaires de L'ingénierie Pétroliere,, Math. et Appl., 22 (1996).   Google Scholar

[22]

M. Karimou Gazibo, Degenerate Convection-Diffusion Equation with a Robin boundary condition,, In F. Ancona et al., 8 (2014), 583.   Google Scholar

[23]

M. Karimou Gazibo, Degenerate parabolic equation with zero flux boundary condition and its approximations,, Preprint available at , ().   Google Scholar

[24]

M. Karimou Gazibo, Études Mathématiques et Numériques des Problèmes Paraboliques Avec Des Conditions Aux Limites,, Thèse de Doctorat Besançon, (2013).   Google Scholar

[25]

S. N. Kruzhkov, First order quasi-linear equations in several independent variables,, Math. USSR Sb., 10 (1970), 217.   Google Scholar

[26]

Y. S. Kwon, Strong traces for degenerate parabolic-hyperbolic equations,, Discrete Contin. Dyn. Syst., 25 (2009), 1275.  doi: 10.3934/dcds.2009.25.1275.  Google Scholar

[27]

M. Maliki and H. Touré, Uniqueness of entropy solutions for nonlinear degenerate parabolic problems,, J. Evol. Equ., 3 (2003), 603.  doi: 10.1007/s00028-003-0105-z.  Google Scholar

[28]

F. Otto, Initial-boundary value problem for a scalar conservation laws,, C. R. Acad. Sci. Paris Sér I Math., 322 (1996), 729.   Google Scholar

[29]

C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equation,, Arch. Ration. Mech. Anal., 163 (2002), 87.  doi: 10.1007/s002050200184.  Google Scholar

[30]

A. Michel and J. Vovelle, Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods,, SIAM J. Numer. Anal., 41 (2003), 2262.  doi: 10.1137/S0036142902406612.  Google Scholar

[31]

E. Yu. Panov, On the theory of generalized entropy solutions of Cauchy problem for a first-order quasilinear equation in the class of locally integrable functions,, Iszvestiya Math., 66 (2002), 1171.  doi: 10.1070/IM2002v066n06ABEH000411.  Google Scholar

[32]

E. Yu. Panov, Existence of strong traces for quasi-solutions of multi-dimensional scalar conservation laws,, J. Hyp. Diff. Equ., 4 (2009), 729.  doi: 10.1142/S0219891607001343.  Google Scholar

[33]

E. Yu. Panov, On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux,, J. Differ. Equ., 247 (2009), 2821.  doi: 10.1016/j.jde.2009.08.022.  Google Scholar

[34]

A. Porretta and J. Vovelle, $L^1$ solutions to first order hyperbolic equations in bounded domains,, Comm. PDEs, 28 (2003), 381.  doi: 10.1081/PDE-120019387.  Google Scholar

[35]

E. Rouvre and G. Gagneux, Formulation forte entropique de lois scalaires hyperboliques-paraboliques dégénérées,, An. Fac. Sci. Toulouse, 10 (2001), 163.  doi: 10.5802/afst.987.  Google Scholar

[36]

T. Tassa, Regularity of weak solutions of the nonlinear Fokker-Planck equation,, Math. Res. Lett., 3 (1996), 475.  doi: 10.4310/MRL.1996.v3.n4.a6.  Google Scholar

[37]

G. Vallet, Dirichlet problem for a degenerated hyperbolic-parabolic equation,, Advance in Math. Sci. Appl., 15 (2005), 423.   Google Scholar

[38]

A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws,, Arch. Ration. Mech. Anal., 160 (2001), 181.  doi: 10.1007/s002050100157.  Google Scholar

[39]

A. I. Vol'pert and S. I. Hudjaev, Cauchy problem for degenerate second order quasilinear parabolic equations,, Math. USSR Sbornik, 78 (1969), 374.   Google Scholar

[40]

J. Vovelle, Convergence of finite volume monotones schemes for scalar conservation laws on bounded domains,, Numer. Math., 90 (2002), 563.  doi: 10.1007/s002110100307.  Google Scholar

show all references

References:
[1]

J. Aleksic and D. Mitrovic, Strong traces for averaged solutions of heterogeneous ultra-parabolic transport equations,, J. Hyperb. Differ. Equ., 10 (2013), 659.  doi: 10.1142/S0219891613500239.  Google Scholar

[2]

K. Ammar, P. Wittbold and J. Carrillo, Scalar conservation laws with general boundary condition and continuous flux function,, J. Differ. Equ., 228 (2006), 111.  doi: 10.1016/j.jde.2006.05.002.  Google Scholar

[3]

B. Andreianov, M. Bendahmane and K. H. Karlsen, Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations,, J. Hyperb. Differ. Equ., 7 (2010), 1.  doi: 10.1142/S0219891610002062.  Google Scholar

[4]

B. Andreianov and F. Bouhsiss, Uniqueness for an elliptic-parabolic problem with Neumann boundary condition,, J. Evol. Equ., 4 (2004), 273.  doi: 10.1007/s00028-004-0143-1.  Google Scholar

[5]

B. Andreianov and M. Karimou Gazibo, Entropy formulation of degenerate parabolic equation with zero-flux boundary condition,, Z. Angew. Math. Phys., 64 (2013), 1471.  doi: 10.1007/s00033-012-0297-6.  Google Scholar

[6]

B. Andreianov and M. Karimou Gazibo, Convergence of finite volume scheme for degenerate parabolic problem with zero flux boundary condition，, Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects, 77 (2014), 303.  doi: 10.1007/978-3-319-05684-5_29.  Google Scholar

[7]

B. Andreianov and K. Shibi, Scalar conservation laws with nonlinear boundary conditions,, C. R. Acad. Paris Ser. I Math., 345 (2007), 431.  doi: 10.1016/j.crma.2007.09.008.  Google Scholar

[8]

B. Andreianov and K. Sbihi, Well-posedness of general boundary-value problems for scalar conservation laws,, Trans. AMS, 367 (2015), 3763.  doi: 10.1090/S0002-9947-2015-05988-1.  Google Scholar

[9]

C. Bardos, A.-Y. LeRoux and J.-C. Nédélec, First order quasilinear equations with boundary conditions,, Comm. PDE, 4 (1979), 1017.  doi: 10.1080/03605307908820117.  Google Scholar

[10]

Ph. Bénilan, Equations D'évolution dans un Espace de Banach Quelconque et Applications,, Thèse d'état, (1972).   Google Scholar

[11]

Ph. Bénilan, M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces,, Preprint book., ().   Google Scholar

[12]

Ph. Bénilan, J. Carrillo and P. Wittbold, Renormalized entropy solutions of scalar conservation laws,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 313.   Google Scholar

[13]

R. Bürger, H. Frid and K.H. Karlsen, On the well-posedness of entropy solution to conservation laws with a zero-flux boundary condition,, J. Math. Anal. Appl., 326 (2007), 108.  doi: 10.1016/j.jmaa.2006.02.072.  Google Scholar

[14]

R. Bürger, H. Frid and K. H. Karlsen, On a free boundary problem for a strongly degenerate quasilinear parabolic equation with an application to a model of presssure filtration,, SIAM J. Math. Anal., 34 (2003), 611.   Google Scholar

[15]

C. Cancès, Th. Gallouët and A. Porretta, Two-phase flows involving capillary barriers in heterogeneous porous media,, Interfaces Free Bound, 11 (2009), 239.  doi: 10.4171/IFB/210.  Google Scholar

[16]

J. Carrillo, Entropy solutions for nonlinear degenerate problems,, Arch. Ration. Mech. Anal., 147 (1999), 269.  doi: 10.1007/s002050050152.  Google Scholar

[17]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws,, Arch. Ration. Mech. Anal., 147 (1999), 89.  doi: 10.1007/s002050050146.  Google Scholar

[18]

R. Colombo and E. Rossi, Rigorous estimates on balance laws in bounded domains,, Acta Mathematica Sci., 35 (2015), 906.  doi: 10.1016/S0252-9602(15)30028-X.  Google Scholar

[19]

F. Dubois and Ph. LeFloch, Boundary condition for nonlinear hyperbolic conservation laws,, J. Differ. Equ., 71 (1988), 93.  doi: 10.1016/0022-0396(88)90040-X.  Google Scholar

[20]

S. Evje and K. H. Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations,, SIAM J. Numer. Anal., 37 (2000), 1838.  doi: 10.1137/S0036142998336138.  Google Scholar

[21]

G. Gagneux and M. Madaune-Tort, Analyse Mathématique de Modeles Nonlinéaires de L'ingénierie Pétroliere,, Math. et Appl., 22 (1996).   Google Scholar

[22]

M. Karimou Gazibo, Degenerate Convection-Diffusion Equation with a Robin boundary condition,, In F. Ancona et al., 8 (2014), 583.   Google Scholar

[23]

M. Karimou Gazibo, Degenerate parabolic equation with zero flux boundary condition and its approximations,, Preprint available at , ().   Google Scholar

[24]

M. Karimou Gazibo, Études Mathématiques et Numériques des Problèmes Paraboliques Avec Des Conditions Aux Limites,, Thèse de Doctorat Besançon, (2013).   Google Scholar

[25]

S. N. Kruzhkov, First order quasi-linear equations in several independent variables,, Math. USSR Sb., 10 (1970), 217.   Google Scholar

[26]

Y. S. Kwon, Strong traces for degenerate parabolic-hyperbolic equations,, Discrete Contin. Dyn. Syst., 25 (2009), 1275.  doi: 10.3934/dcds.2009.25.1275.  Google Scholar

[27]

M. Maliki and H. Touré, Uniqueness of entropy solutions for nonlinear degenerate parabolic problems,, J. Evol. Equ., 3 (2003), 603.  doi: 10.1007/s00028-003-0105-z.  Google Scholar

[28]

F. Otto, Initial-boundary value problem for a scalar conservation laws,, C. R. Acad. Sci. Paris Sér I Math., 322 (1996), 729.   Google Scholar

[29]

C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equation,, Arch. Ration. Mech. Anal., 163 (2002), 87.  doi: 10.1007/s002050200184.  Google Scholar

[30]

A. Michel and J. Vovelle, Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods,, SIAM J. Numer. Anal., 41 (2003), 2262.  doi: 10.1137/S0036142902406612.  Google Scholar

[31]

E. Yu. Panov, On the theory of generalized entropy solutions of Cauchy problem for a first-order quasilinear equation in the class of locally integrable functions,, Iszvestiya Math., 66 (2002), 1171.  doi: 10.1070/IM2002v066n06ABEH000411.  Google Scholar

[32]

E. Yu. Panov, Existence of strong traces for quasi-solutions of multi-dimensional scalar conservation laws,, J. Hyp. Diff. Equ., 4 (2009), 729.  doi: 10.1142/S0219891607001343.  Google Scholar

[33]

E. Yu. Panov, On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux,, J. Differ. Equ., 247 (2009), 2821.  doi: 10.1016/j.jde.2009.08.022.  Google Scholar

[34]

A. Porretta and J. Vovelle, $L^1$ solutions to first order hyperbolic equations in bounded domains,, Comm. PDEs, 28 (2003), 381.  doi: 10.1081/PDE-120019387.  Google Scholar

[35]

E. Rouvre and G. Gagneux, Formulation forte entropique de lois scalaires hyperboliques-paraboliques dégénérées,, An. Fac. Sci. Toulouse, 10 (2001), 163.  doi: 10.5802/afst.987.  Google Scholar

[36]

T. Tassa, Regularity of weak solutions of the nonlinear Fokker-Planck equation,, Math. Res. Lett., 3 (1996), 475.  doi: 10.4310/MRL.1996.v3.n4.a6.  Google Scholar

[37]

G. Vallet, Dirichlet problem for a degenerated hyperbolic-parabolic equation,, Advance in Math. Sci. Appl., 15 (2005), 423.   Google Scholar

[38]

A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws,, Arch. Ration. Mech. Anal., 160 (2001), 181.  doi: 10.1007/s002050100157.  Google Scholar

[39]

A. I. Vol'pert and S. I. Hudjaev, Cauchy problem for degenerate second order quasilinear parabolic equations,, Math. USSR Sbornik, 78 (1969), 374.   Google Scholar

[40]

J. Vovelle, Convergence of finite volume monotones schemes for scalar conservation laws on bounded domains,, Numer. Math., 90 (2002), 563.  doi: 10.1007/s002110100307.  Google Scholar

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